Misa-Math 1

Revision en61, by Misa-Misa, 2023-11-21 07:54:21

1

Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $$$\frac{m}{n},$$$ where $$$m$$$ and $$$n$$$ are relatively prime positive integers. Find $$$m+n.$$$

Answer
Solution

2

A plane contains $$$40$$$ lines, no $$$2$$$ of which are parallel. Suppose there are $$$3$$$ points where exactly $$$3$$$ lines intersect, $$$4$$$ points where exactly $$$4$$$ lines intersect, $$$5$$$ points where exactly $$$5$$$ lines intersect, $$$6$$$ points where exactly $$$6$$$ lines intersect, and no points where more than $$$6$$$ lines intersect. Find the number of points where exactly $$$2$$$ lines intersect.

Answer
Solution

3

The sum of all positive integers $$$m$$$ for which $$$\tfrac{13!}{m}$$$ is a perfect square can be written as $$$2^{a}3^{b}5^{c}7^{d}11^{e}13^{f}$$$, where $$$a, b, c, d, e,$$$ and $$$f$$$ are positive integers. Find $$$a+b+c+d+e+f$$$.

Answer
Solution

4

There exists a unique positive integer $$$a$$$ for which the sum $$$[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor]$$$ is an integer strictly between $$$-1000$$$ and $$$1000$$$. For that unique $$$a$$$, find $$$a+U$$$.

(Note that $$$\lfloor x\rfloor$$$ denotes the greatest integer that is less than or equal to $$$x$$$.)

Answer

5

Consider an $$$n$$$-by-$$$n$$$ board of unit squares for some odd positive integer $$$n$$$. We say that a collection $$$C$$$ of identical dominoes is a maximal grid-aligned configuration on the board if $$$C$$$ consists of $$$(n^2-1)/2$$$ dominoes where each domino covers exactly two neighboring squares and the dominoes don't overlap: $$$C$$$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $$$k(C)$$$ be the number of distinct maximal grid-aligned configurations obtainable from $$$C$$$ by repeatedly sliding dominoes. Find the maximum value of $$$k(C)$$$ as a function of $$$n$$$.

Answer




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en61 English Misa-Misa 2023-11-21 07:54:21 74
en60 English Misa-Misa 2023-11-21 07:01:54 90
en59 English Misa-Misa 2023-11-21 06:47:13 2 Tiny change: 'n+1)^{2}}{2}$\n</spoi' -> 'n+1)^{2}}{4}$\n</spoi'
en58 English Misa-Misa 2023-11-21 05:33:48 0 (published)
en57 English Misa-Misa 2023-11-21 05:11:05 3
en56 English Misa-Misa 2023-11-21 05:10:16 12 Tiny change: 'nswer">\n$(n+1)^{2} / 2$\n</spoil' -> 'nswer">\n$\frac{(n+1)^{2}}{2}$\n</spoil'
en55 English Misa-Misa 2023-11-21 05:09:28 5 Tiny change: '">\n$(n+1)_2 / 2$\n</s' -> '">\n$(n+1)^{2} / 2$\n</s'
en54 English Misa-Misa 2023-11-21 05:08:45 919
en53 English Misa-Misa 2023-11-11 05:25:24 1 Tiny change: 'o $x$.)<br\n\n<spoil' -> 'o $x$.)<br>\n\n<spoil'
en52 English Misa-Misa 2023-11-10 14:35:07 259
en51 English Misa-Misa 2023-11-10 14:34:45 261
en50 English Misa-Misa 2023-11-10 06:53:02 1217
en49 English Misa-Misa 2023-11-10 06:51:09 9
en48 English Misa-Misa 2023-11-10 06:50:02 1669
en47 English Misa-Misa 2023-11-02 11:06:57 1
en46 English Misa-Misa 2023-11-02 11:06:28 17 Tiny change: 'oiler>\n\n\n<spoiler' -> 'oiler>\n\n<br>\n<br>\n<br>\n<spoiler'
en45 English Misa-Misa 2023-11-02 08:53:05 12
en44 English Misa-Misa 2023-11-02 08:52:20 130
en43 English Misa-Misa 2023-11-02 08:50:13 3
en42 English Misa-Misa 2023-11-02 08:49:02 156
en41 English Misa-Misa 2023-11-02 08:46:01 159
en40 English Misa-Misa 2023-11-02 08:42:29 6
en39 English Misa-Misa 2023-11-02 08:41:53 69
en38 English Misa-Misa 2023-11-02 08:40:14 417
en37 English Misa-Misa 2023-11-02 08:35:41 107
en36 English Misa-Misa 2023-11-02 08:28:37 41
en35 English Misa-Misa 2023-11-02 08:27:43 2 Tiny change: 'n}{x} + \farc{n}{x^{2}' -> 'n}{x} + \frac{n}{x^{2}'
en34 English Misa-Misa 2023-11-02 08:27:17 3
en33 English Misa-Misa 2023-11-02 08:26:39 2 Tiny change: ' $\lfloor{frac{n}{x}' -> ' $\lfloor frac{n}{x}'
en32 English Misa-Misa 2023-11-02 08:26:09 47
en31 English Misa-Misa 2023-11-02 08:22:53 14 Tiny change: 'n!$ = $\lceil{x}\rceil$\n\n</spo' -> 'n!$ = $\lfloor{x}\rfloor$\n\n</spo'
en30 English Misa-Misa 2023-11-02 08:21:54 58
en29 English Misa-Misa 2023-11-02 08:17:00 0
en28 English Misa-Misa 2023-11-02 08:17:00 388
en27 English Misa-Misa 2023-11-02 07:29:46 22
en26 English Misa-Misa 2023-11-02 07:28:47 55
en25 English Misa-Misa 2023-11-02 07:27:34 710
en24 English Misa-Misa 2023-11-02 07:20:08 398
en23 English Misa-Misa 2023-11-02 07:01:20 19
en22 English Misa-Misa 2023-11-02 07:00:49 142
en21 English Misa-Misa 2023-11-02 06:59:50 22
en20 English Misa-Misa 2023-11-02 06:59:04 16 Tiny change: 'om{14}{5}}= \frac{48}{143}$\n</spoi' -> 'om{14}{5}}$\n</spoi'
en19 English Misa-Misa 2023-11-02 06:58:35 22
en18 English Misa-Misa 2023-11-02 06:57:29 12
en17 English Misa-Misa 2023-11-02 06:56:48 68
en16 English Misa-Misa 2023-11-02 06:54:37 67
en15 English Misa-Misa 2023-11-02 06:53:37 70
en14 English Misa-Misa 2023-11-02 06:52:56 4 Tiny change: ' = $\frac{!14}{!5}$\nThere ' -> ' = $\frac{14!}{5!}$\nThere '
en13 English Misa-Misa 2023-11-02 06:52:39 33
en12 English Misa-Misa 2023-11-02 06:51:33 98
en11 English Misa-Misa 2023-11-02 06:50:45 147
en10 English Misa-Misa 2023-11-02 06:28:38 4 Tiny change: 'ng skills.\nTarget a' -> 'ng skills.<br>\nTarget a'
en9 English Misa-Misa 2023-11-02 06:28:24 39 Tiny change: ' skills.\n</spoil' -> ' skills.\nTarget audience : Experts and below.\n</spoil'
en8 English Misa-Misa 2023-11-02 06:27:34 11 Tiny change: 'm+n.$ \n\n\n<spoiler' -> 'm+n.$ \n\n<br>\n<br>\n<spoiler'
en7 English Misa-Misa 2023-11-02 06:27:11 135
en6 English Misa-Misa 2023-11-02 06:26:17 9 Tiny change: 'ath">\n```c++\nI plan t' -> 'ath">\n```python\nI plan t'
en5 English Misa-Misa 2023-11-02 06:25:51 19
en4 English Misa-Misa 2023-11-02 06:25:14 59
en3 English Misa-Misa 2023-11-02 06:23:58 177
en2 English Misa-Misa 2023-11-02 06:22:43 167
en1 English Misa-Misa 2023-11-02 06:18:27 249 Initial revision (saved to drafts)